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02_averages_final |
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Regression of \(y\) and \(\hat y\) on \(\mathbf{1}\). |
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02_averages_yhat_decomposed |
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Regression of \(y\) on \(Lin(\mathbf{1},x)\) and decomposition of \(\hat y\) into a sum of \(\hat\beta_1 \mathbf{1}\) and \(\hat \beta_2 x\). |
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02_basic_projection |
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Vector \(y\) projected onto vector \(x\). |
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02_correlation_constant_centered_variables |
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Centred vectors \(x^c\) and \(y^c\). |
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02_correlation_constant_proof |
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Proof of \(sCorr(x + \alpha \mathbf{1}, y) = sCorr(x,y)\). |
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02_cramers_rule |
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OLS formula illustrated in \(\mathbb{R}^k\). |
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02_detremination_coefficient |
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Determination coefficient as squared \(\cos \varphi\) where \(a\) stands for \(\sqrt{RSS}\), \(b\) — \(\sqrt{TSS}\), \(c\) — \(\sqrt{ESS}\). |
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02_duality_final |
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New residuals translated to the origin of the unit circle. |
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02_duality_first_residuals |
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Residuals \(\hat{u}_1\) and \(\hat{u}_2\). |
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02_duality_first_residuals_translated |
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Residuals translated to the orgin of the unit circle. |
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02_duality_inversion |
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Example of inversion for vector \(a\). |
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02_duality_new_regressors |
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Regressors \(v_1\), \(v_2\) obtained from inversion of the residuals \(\hat{u}_1\), \(\hat{u}_2\). |
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02_duality_new_residuals |
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Regressions of \(v_1\) onto \(v_2\) and of \(v_2\) onto \(v_1\). |
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02_duality_original_regressors |
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Original regressors in unit circle. |
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02_fwl_v1_cleansed_variables |
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FWL: regression of \(y\) on \(z\) and of \(x\) on \(z\). |
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02_fwl_v1_final |
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FWL: point A stands for the origin, B — \(\hat\gamma z\), C — \(x\), D — \(\hat\alpha z\), E — intersection of vector \(x\) and line parallel to \(\tilde x\), F — \(\hat\beta_1^{A} x\), G — \(\hat\beta_1^{B} \tilde{x}\). |
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02_fwl_v1_final_lin |
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FWL: \(Lin(x,z)\). |
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02_fwl_v1_translation |
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FWL: translation of \(\tilde{x}\). |
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02_fwl_v1_yhat_decomposed |
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FWL: regression of \(y\) on \(Lin(x,z)\). |
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02_fwl_v1_yhat_decomposed_lin |
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FWL: \(Lin(x, z)\). |
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02_fwl_v2_cleansed_regression |
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FWL: “cleansed” \(\tilde y\) regressed on “cleansed” \(\tilde{x}\). |
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02_fwl_v2_cleansed_variables |
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FWL: “cleansed” variables \(\tilde x\) and \(\tilde y\). |
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02_fwl_v2_final |
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Alternative proof for the Frisch-Waugh-Lovell theorem. |
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02_fwl_v2_similar_triangles |
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FWL: similar triangles \(\bigtriangleup ABC \sim \bigtriangleup EDC\). |
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02_gmt |
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Gauss-Markov theorem for the case of three regressors. |
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02_instr |
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Geometry of instrumental variables. \(A\) stands for \(\hat \beta_{IV} \hat x\), \(B\) — \(\hat x\), \(C\) — \(x\), \(D\) — \(\hat \beta_{IV} x\). |
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02_proxy |
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Geometry of proxy variables. |
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02_rss_ess_tss_final |
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Illustration of the equality \((\sqrt{RSS})^2 + (\sqrt{ESS})^2 = (\sqrt{TSS})^2\) where \(a\) stands for \(\sqrt{RSS}\), \(b\) — \(\sqrt{TSS}\), \(c\) — \(\sqrt{ESS}\). |
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02_rss_ess_tss_sqr_ess |
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Illustartion of ESS. |
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02_rss_ess_tss_sqr_rss |
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Illustration of RSS. |
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02_rss_ess_tss_sqr_tss_ess |
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Total sum of squares and residual sum of squares. |
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02_rss_ess_tss_yhat |
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Residual sum of squares. |
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02_simple_regression_coefficient_basic |
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Starting pictutre for regression illustrations. |
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02_simple_regression_coefficient_centred_variables |
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“Centred” \(x\) and \(y\), i.e., projected onto \(Lin^{\perp}(\mathbf{1})\). |
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02_simple_regression_coefficient_negative |
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Proof of \(Corr(y, \hat y) = sign(\hat\beta_2)Corr(y, x)\) for the case of \(\beta_2 < 0\). |
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02_simple_regression_coefficient_yhat_projected |
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“Centred” \(\hat y\), i.e., projected onto \(Lin^{\perp}(\mathbf{1})\). |
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02_uncorrelated_regressors |
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Illustration of uncorrelated regresssors. |